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    <title>欧氏空间</title>
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<h2>向量空间</h2>

<h3>内积 `(: alpha, beta :)`</h3>

<p>内积定义为具有正定性的对称双线性函数:</p>

<ol>
	<li>`(: alpha, alpha :) ge 0`, 等号成立 `iff alpha = 0`</li>
	<li>`(: alpha, beta :) = (: beta, alpha :)`</li>
	<li>`k (: alpha, beta :) = (: k alpha, beta :)`</li>
	<li>`(: alpha+beta, gamma :) = (: alpha, gamma :) + (: beta, gamma :)`
	</li>
</ol>

<p>设 `X, Y in RR^n`, 则
	<span class="formula">
		`(:X, Y:) = sum_(i=1)^n x_i y_i in RR`.
	</span>
</p>

<p>设 `X in RR^3`, 则
	<span class="formula">
		`X = (:i, X:)i + (:j, X:)j + (:k, X:)k`.
	</span>
</p>

<p>称 `alpha, beta` <b>正交</b> (或垂直), 如果 `(:alpha, beta:) = 0`.</p>

<h3>模（长度） `|alpha| = sqrt((: alpha, alpha :))`</h3>

<ol>
	<li>`|alpha| ge 0`, 等号成立 `iff alpha = 0`</li>
	<li>`|k alpha| = |k| |alpha|`</li>
	<li>`|alpha + beta| le |alpha| + |beta|`</li>
</ol>

<p>设 `X in RR^n`, 则
	<span class="formula">
		`|X| = sqrt(sum_(i=1)^n x_i^2)`.
	</span>
</p>

<h3>外积 `alpha ^^ beta`</h3>

<p>外积是一向量, 模等于两向量所夹的平行四边形的面积, 方向满足右手定则.</p>

<ol>
	<li>`alpha ^^ beta = -beta ^^ alpha`</li>
	<li>`(alpha+beta) ^^ gamma = alpha ^^ gamma + beta ^^ gamma`</li>
	<li>`gamma ^^ (alpha+beta) = gamma ^^ alpha + gamma ^^ beta`</li>
	<li>`k(alpha ^^ beta) = (k alpha) ^^ beta = alpha ^^(k beta)`</li>
</ol>

<p>设 `X, Y in RR^3`, 则
	<span class="formula">
		`X ^^ Y = |
			i, j, k;
			x_1, x_2, x_3;
			y_1, y_2, y_3;
		| in RR^3`.
	</span>
</p>

<p>`alpha, beta` 线性相关 (共线) 当且仅当 `alpha ^^ beta = bb 0`.</p>

<h3>混合积 `(alpha, beta, gamma) = (:alpha ^^ beta, gamma:) in RR^3`</h3>

<p>`(alpha, beta, gamma) = (beta, gamma, alpha) = (gamma, alpha, beta)`</p>

<p>设 `X, Y, Z in RR^3`, 则
	<span class="formula">
		`(X, Y, Z) = (:|
			i, j, k;
			x_1, x_2, x_3;
			y_1, y_2, y_3;
		|, Z:) = |
			x_1, x_2, x_3;
			y_1, y_2, y_3;
			z_1, z_2, z_3;
		|.`
	</span>
</p>

<p>`alpha, beta, gamma` 线性相关 (共面) 当且仅当 `(alpha, beta, gamma) = 0`.</p>

<h3>恒等式 / 不等式</h3>

<ol>
	<li>(由定义或由行列式性质) `(:alpha ^^ beta, alpha:) = (:alpha ^^
		beta, beta:) = 0`;</li>
	<li>(平行四边形等式) `|alpha+beta|^2 + |alpha-beta|^2 =
		2(|alpha|^2+|beta|^2)`;</li>
	<li>(二重外积) `alpha ^^ (beta ^^ gamma) = |
		beta, gamma;
		(:alpha,beta:),(:alpha,gamma:);
	|`;</li>
	<li>(推论, 注意联系几何意义) 若 `(:alpha, beta:) = (:alpha, gamma:) =
		0`, 则 `alpha ^^ (beta ^^ gamma) = bb 0`.
	</li>
	<li>(Lagrange) `(:alpha ^^ beta, gamma ^^ delta:) = |
		(:alpha, gamma:), (:alpha, delta:);
		(:beta, gamma:), (:beta, delta:);
	|`;</li>
	<li>(Cauchy) `|(:alpha, beta:)| le |alpha||beta|`, 等号成立当且仅当
		`alpha, beta` 线性相关;
	</li>
</ol>

<p class="remark">Lagrange 恒等式的证明:
	<span class="formula">
		左 = `(:beta ^^ (gamma ^^ delta), alpha:)
		= (:|gamma, delta; (:beta, gamma:), (:beta, delta:)|, alpha:)`
		= 右.
	</span>
	由 Lagrange 恒等式推出外积公式:
	<span class="formula">
		` X ^^ Y
		= sum (:X ^^ Y, epsi_i:) epsi_i
		= sum(:X ^^ Y, epsi_j ^^ epsi_k:) epsi_i`
		`= sum |
			(:X, epsi_j:), (:X, epsi_k:);
			(:Y, epsi_j:), (:Y, epsi_k:)| epsi_i
		= | i, j, k;
		(:X, i:), (:X, j:), (:X, k:);
		(:Y, i:), (:Y, j:), (:Y, k:)|`.
	</span>
</p>

<h2>向量分析</h2>

<h3>向量/矩阵的导数</h3>

<p>定义 `alpha(t) = (a_1(t), a_2(t), a_3(t))` 的导数:
	<span class="formula">
		`d/dt alpha(t) = ((da_1)/dt, (da_2)/dt, (da_3)/dt)`.
	</span>
</p>

<p>设 `lambda = lambda(t)` 是数值函数, 有:</p>

<ol>
	<li>`d/dt (alpha+beta) = (d alpha)/dt + (d beta)/dt`</li>
	<li>`d/dt (lambda alpha) = (d lambda)/dt alpha + lambda (d alpha)/dt`</li>
	<li>`d/dt (:alpha, beta:) = (: (d alpha)/dt, beta :) + (:alpha, (d
		beta)/dt:)`</li>
	<li>`d/dt (alpha ^^ beta) = (d alpha)/dt ^^ beta + alpha ^^ (d
		beta)/dt`</li>
	<li>`d/dt (alpha, beta, gamma) = ((d alpha)/dt, beta, gamma) + (alpha,
		(d beta)/dt, gamma) + (alpha, beta, (d gamma)/dt)`</li>
	<li>A(t), B(t) 为矩阵. `d/dt (A(t)B(t)) = (dA)/dt B + A (dB)/dt`</li>
</ol>

<p class="remark">
	1, 2 验证分量即可. 3 由 `(:alpha, beta:) = sum a_i b_i` 可证. 4
	由行列式求导的法则可得. 5 由 3, 4 可得.
</p> 

<h3>Nabla 算子 `grad = (del/(del x), del/(del y), del/(del z))`</h3>

<p>
	设 `f(x, y, z)` 为数量函数, `F(x, y, z) = (P(x, y, z), Q(x, y, z),
	R(x, y, z))` 为向量函数 (或称向量场).
</p>

<p>
	梯度 `"grad" f =  grad f`, 散度 `"div" F = (:grad, F:)`, 旋度
	`"rot" F = grad ^^ F`.
</p>

<ol>
	<li>`(:grad, F+G:) = (:grad, F:) + (:grad, G:)`</li>
	<li>`(:grad, fF:) = f(:grad, F:) + (:grad f, F:)`</li>
	<li>`grad ^^ (F+G) = grad ^^ F + grad ^^ G`</li>
	<li>`grad ^^ (fF) = f(grad ^^ F) + (grad f) ^^ F`</li>
	<li>(梯度场是无旋场) `grad ^^ (grad f) = bb"0"`</li>
	<li>(旋度场是无源场) `(: grad, grad ^^ F:) = 0`</li>
</ol>

<p class="remark">
	2, 4 相当于将 `grad` 分配到 f 与 F. 5 假定混合偏导相等, 故
	<span class="formula">
		`|
			i, j, k;
			del/(del x), del/(del y), del/(del z);
			(del f)/(del x), (del f)/(del y), (del f)/(del z);
		| = bb"0"`.
	</span>
	6 同样假定混合偏导相等, 有
	<span class="formula">
		`|
			del/(del x), del/(del y), del/(del z);
			del/(del x), del/(del y), del/(del z);
			P, Q, R;
		| = 0`.
	</span>
</p>

<h2>合同变换 (保距变换)</h2>

<p>具有形式
	<span class="formula">
		`cc"T"(X) = cc"T" X + X_0`, 其中 `cc"T"` 为正交实矩阵.
	</span>
</p>

<p class="theorem">
	设 `T` 为 `n` 阶正交阵, `t_i` 是其第 `i` 列 (行), 有
	<span class="formula">
		`(:t_i, t_j:) = delta_(ij)`.
	</span>
</p>

<p class="corollary">
	在上述定理中特别取 `n = 3` 时, `t_1 ^^ t_2 = "sgn"|T| t_3`.
</p>

<p class="proof">
	不妨设 `|T| = 1`,
	因为 `(:t_1 ^^ t_2, t_3:) = |T| = 1`, 但
	<span class="formula">
		`|t_1 ^^ t_2|^2 = (:t_1 ^^ t_2, t_1 ^^ t_2:) = |
			(:t_1, t_1:), (:t_1, t_2:);
			(:t_2, t_1:), (:t_2, t_2:);
		| = 1`, <br/>
		`|t_3|^2 = (:t_3, t_3:) = 1`.
	</span>
	由 Cauchy 不等式知 `t_1 ^^ t_2 = t_3`.
</p>

<p class="theorem">
	设 `T` 为正交阵, 则 `T (alpha ^^ beta) = T alpha ^^ T beta`.
</p>

<p class="proof">
	记 `t_i` 为 T 的第 i 行,
	<span class="formula">
		左 = `sum (:t_i, alpha ^^ beta:)
		= sum (:t_j ^^ t_k, alpha ^^ beta:) epsi_i`
		`= sum |
			(:t_j, alpha:), (:t_j, beta:);
			(:t_k, alpha:), (:t_k, beta:); | epsi_i
		= |
			i, j, k;
			(:alpha, t_1:), (:alpha, t_2:), (:alpha, t_3:);
			(:beta, t_1:), (:beta, t_2:), (:beta, t_3:);
		| = T alpha ^^ T beta`.
	</span>
</p>

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